Finite Crystal Time-Frequency Bridge

Abstract

Distinct finite crystal modes are exactly reconstructible from an equally long scalar time window.

Theorem 1.1 (Separated modes are recovered from time samples).

$$\forall omega,\mathrm{Injective}(omega)\Rightarrow \mathrm{Injective}(\mathrm{firstCrystalTimeWindow}(omega)).$$

Proof. Machine-checked in Lean as D5/S3/ObserverMemory/FourierFibers/FiniteCrystalTimeFrequencyBridge.first_crystal_time_window_injective (✓ std3). ∎

Source. Repository-derived.

Commentary.

For finitely many distinct modal multipliers, the first matching number of scalar time samples uniquely recovers all modal amplitudes.

This is a finite diagonal spectral realization of Vandermonde tomography. It does not construct an infinite Bloch bundle or identify the sampling index with physical time.

References

• Truth anchor: D5/S3/ObserverMemory/FourierFibers/FiniteCrystalTimeFrequencyBridge.first_crystal_time_window_injective
• Dependency: D5/S3/Analytic/GoldenTomography/FiniteVandermondeTomography